Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

Abstract

We consider the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long wavelength fluctuations in a broad class of one-dimensional substitution tilings. We present a simple argument that predicts the exponent $\alpha$ governing the scaling of Fourier intensities at small wavenumbers, tilings with $\alpha>0$ being hyperuniform, and confirm with numerical computations that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra, and limit-periodic tilings. Tilings with quasiperiodic or singular continuous spectra can be constructed with $\alpha$ arbitrarily close to any given value between $-1$ and $3$. Limit-periodic tilings can be constructed with $\alpha$ between $-1$ and $1$ or with Fourier intensities that approach zero faster than any power law.